Problem: Howard is designing a chair swing ride. The swing ropes are $4$ meters long, and in full swing they tilt in an angle of $23^\circ$. Howard wants the chairs to be $3.5$ meters above the ground in full swing. How tall should the pole of the swing ride be? Round your final answer to the nearest hundredth.
The strategy Model the situation as a right triangle. Determine the appropriate trigonometric ratio in order to find the missing side. Form an equation and solve for the missing side. Calculate the final result and round. Modeling as a right triangle This situation can be modeled by the following right triangle that is located $3.5\text{ m}$ above the ground. The hypotenuse is $4\text{ m}$ and the top angle is $23^\circ$. We are asked to find the length of the pole, so we need to find the height of the triangle (and add $3.5\text{ m}$ to it). ${23^{\circ}}$ $4$ $?$ $3.5$ Determining the appropriate trigonometric ratio We are given the measure of an angle and the length of the $C{\text{hypotenuse}}$. We are asked to find the side ${\text{adjacent}}$ to the given angle. The appropriate trigonometric ratio is therefore the $\text{cosine}$. Forming an equation and solving Denoting the missing side by $x$, we obtain the equation $\cos(23^\circ)=\dfrac{x}{4}$. Solving the equation, we get $x=4\cdot\cos(23^\circ)$. Evaluating this result in the calculator and rounding to the nearest hundredth, we get $x=3.68\text{ m}$. Remember that this is the part of the pole that is above $3.5$ meters. Therefore, the total length of the pole should be $3.5+3.68=7.18\text{ m}$ Summary The pole of the swing should be $7.18$ meters tall.